Turning the proposition around, why not have regular banks, even the Bank of England. Issue an official GBPCoin, or a banking framework that does the same thing. Have whoever needs stablecoin use that.

That is, stablecoins are basically USD accounts, just with the actual bank receipts (the coins) circulating. Make that a form of deposit account, and regulate and insure it like any other deposit account.

It presents some questions that aren't answered in advance, but I don't think it's impossible to do this right. In some sense, a stablecoin system is easier to understand.